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The Little q-Jacobi Functions of Complex Order

  • Kevin W. J. Kadell
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Part of the Developments in Mathematics book series (DEVM, volume 13)

Abstract

We use Ismail's argument and an elementary combinatorial identity to prove the q-binomial theorem, the symmetry of the Rogers-Fine function, Ramanujan's 1ψ1 sum, and Heine's q-Gauss sum and give many other proofs of these results. We prove a special case of Heine's 2ϕ1 transformation and write Ramanujan's 1ψ1 sum as the nonterminating q-Chu-Vandermonde sum. We show that the q-SaalschÜtz and q-Chu-Vandermonde sums are equivalent to the evaluations of certain moments and to the orthogonality of the little q-Jacobi polynomials; hence the q-Chu-Vandermonde sum implies the q-Saalschütz sum. We extend the little q-Jacobi polynomials naturally to the little q-Jacobi functions of complex order. We show that the nonterminating q-Saalschütz and q-Chu-Vandermonde sums are equivalent to the evaluations of certain moments and, using the Liouville-Ismail argument, to two orthogonality relations. We show that the nonterminating q-Chu-Vandermonde sum implies the nonterminating q-Saalschütz sum.

Keywords

little q-Jacobi polynomials basic hypergeometric functions bilateral basic hypergeometric functions q-binomial theorem Rogers-Fine symmetric function Ramanujan's 1ψ1 sum Heine's q-Gauss sum and 2ϕ1 transformation q-Chu-Vandermonde sum q-Saalschütz sum nonterminating sum and Macdonald Bidenharn-Louck Heckman-Opdam Koornwinder and Sahi-Knop polynomials 

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© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Kevin W. J. Kadell
    • 1
  1. 1.Department of Mathematics and StatisticsArizona State UniversityTempe

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